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Inspired by the question

http://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve

and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth projective connected) curves of genus $3$ contains a (smooth projective connected) curve of genus $2$.

The existence of such a genus two curve is (Edit: stronger) than the existence of a surface $S$, a genus two curve $C$ and a smooth projective non-isotrivial morphism $S\to C$ whose fibres are genus three curves.

If the answer is positive, how explicit can our answer be made? I'm already aware of the fact that $M_g$ contains a complete curve for all $g\geq 3$. For instance, in the paper by Chris Zaal

http://dare.uva.nl/document/38546

many curves of some genus (I think 513) are shown to embed into $M_3$.

Of course, by Shafarevich' conjecture, if $K(C)$ denotes the function field of $C$, there are only finitely many $K(C)$-isomorphism classes of genus three curves over $K(C)$ with good reduction over $C$. I'm asking whether there exists some genus two curve $C$ such that there exists a genus three curve over $K(C)$ with good reduction over $C$.

Edit: the arithmetic analogue also has a negative answer.

The latter (weaker) phrasing of my question allows us to formulate an arithmetic analogue of the above question. (I know that I'm considering function fields over $\mathbf{C}$ and that some of you might argue function fields over $\mathbf{F}_p$ are a better analogue of number fields.) This arithmetic analogue reads as follows. There exists a number field of "genus two" such that there exist a genus three curve over $K$ with good reduction over the ring of integers of $K$. Here a number field of "genus two" should be a number field of absolute discriminant $e^2$. I'll take this to mean discriminant at most $8$.

Arithmetic analogue. (Abrashkin-Fontaine) There do not exist non-zero smooth abelian schemes over the ring of integers of a number field of absolute discriminant at most 8.

There are many related questions I'd also like to ask. For example, what is the minimal $g$ such that $M_g$ contains a genus two curve? Or, what is the minimal $g$ such that $M_3$ contains a genus $g$ curve? And, finally, is there an example of a complete curve in $M_g$ which is defined over $\overline{\mathbf{Q}}$? (Edit: The answer to the last question is positive. This is explained in the comments below.)

edited Nov 6 at 19:18

Ariyan Javanpeykar

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Setting aside the fact that $M_g$ is a stack and not a scheme, non-isotriviality is also too weak: it only says that the map from $C$ to the stack is non-constant, not that it is a closed immersion (or even just injective on geometric points). Also, concerning the question at the very end, by "spreading out and specialization" we see that there are complete curves of a given genus inside $M_g$ over $\overline{\mathbf{Q}}$ if this holds over one algebraically closed field of char. 0 (such as $\mathbf{C}$), or maybe "an example" meant to include "explicit"? – xbnv Nov 5 at 15:26

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But you're claiming that you're "already aware of the fact that $M_g$ contains a complete curve for all $g \geq 2$". Should that be $>$ instead of $\geq$ then? – René Pannekoek Nov 5 at 19:42

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@Will Sawin. The locus of hyperelliptic curves (or more generally cyclic covers of prime degree in $M_g$) is affine. This is a result of Gonzalez Diez (1991); see "Loci of curves which are prime Galois coverings of P^1". Probably the result for hyperelliptic curves is more classical. – Ariyan Javanpeykar Nov 5 at 23:10

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@Will Sawin (continued). So there is no complete curve contained in the locus of hyperelliptic curves of $M_3$. Moreover, the generic fibre of $S\to C$ has to be a smooth quartic. In fact, if it were hyperelliptic, all fibres would be hyperelliptic because $S\to C$ is smooth. – Ariyan Javanpeykar Nov 5 at 23:13

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@Ariyan. Yes, the result for hyperelliptic curves is very classical. There is an isomorphism $H_g \cong M_{0,2g+2}/\mathbb{S}_{2g+2}$, and $M_{0,n}$ is obviously affine (it is the complement of an arrangement of hyperplanes in $\mathbb{A}^{n-3}$). – Dan Petersen Nov 6 at 6:29

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Jim Bryan · Accepted Answer · 2012-11-06 15:48:20Z UTC

There does not exist a map of a smooth complete genus 2 curve to $M_3$.

Such a map would give rise to a surface $S$ (of general type) which violates the Bogomolov-Miyaoka-Yau inequality $c_1(S)^2 \leq 3c_2(S)$. This inequality is equivalent to $3\sigma (S) \leq e(S)$ where $\sigma$ and $e$ are the signature and topological euler characteristic of the surface. The euler characteristic of this surface is 8 (since it is multiplicative for fiber bundles) and by the index theorem, the signature is given by 4 times the integral of $\lambda_1$ over the curve in $M_3$. Since $\lambda_1$ is ample on $M_3$, $\sigma$ must be positive and divisible by 4.

This argument is due to Dieter Kotschick in his paper "Signatures, Monopoles, and Mapping Class Groups" (MRL vol 5, 1998).

Does the moduli space of genus three curves contain a complete genus two curve

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